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Putting Auction Theory to Work P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610-FM CB610-Milgrom-v3 October 27, 2003 16:27 PUTTING AUCTION THEORY TO WORK This book provides a comprehensive introduction to modern auction theory and its important new applications. It is written by a leading economic theorist whose suggestions guided the creation of the new spectrum auction designs. Aimed at graduate students and professionals in economics, the book gives the most up-to-date treatments of both traditional theories of “optimal auctions” and newer theories of multi-unit auctions and package auctions, and shows by example how these theories are used. The analysis explores the limitations of prominent older designs, such as the Vickrey auction design, and evaluates the practical responses to those limitations. It explores the tension between the traditional theory of auctions with a fixed set of bidders, in which the seller seeks to squeeze as much revenue as possible from the fixed set, and the theory of auctions with endogenous entry, in which bidder profits must be respected to encourage participation. It shows how seemingly different auction designs can lead to nearly identical outcomes if the participating bidders are the same – a finding that focuses attention on (1) attracting bidders and (2) mini- mizing the cost of running the auction and bidding in it. It shows how new auc- tion designs can accommodate complicated procurement settings and sales with many interrelated goods. Paul Milgrom is Leonard and Shirley Ely Professor of Humanities and Social Sciences and Professor of Economics at Stanford University.He has also taught at Harvard University and MIT. A Fellow of the American Academy of Arts and Sciences and the Econometric Society, Professor Milgrom has served on the editorial boards of the American Economic Review, Ecometrica, the Journal of Economic Theory, the Journal of Economics and Management Strategy, and Games and Economic Behavior. He is coauthor with John Roberts of the 1992 landmark text Economics, Organization, and Management. Professor Milgrom’s research has been published in the leading journals in economics, including the American Economic Review, Econometrica, the Journal of Political Economy, the Quarterly Journal of Economics, the Journal of Economic Theory, the Journal of Economic Perspectives, and the Journal of MathematicalEconomics.Hiscurrentresearchinterestsareinincentivetheory, planning, and auction market design. Professor Milgrom is internationally known for his work in spectrum auction designs. i P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610-FM CB610-Milgrom-v3 October 27, 2003 16:27 ii P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610-FM CB610-Milgrom-v3 October 27, 2003 16:27 CHURCHILL LECTURES IN ECONOMICS The Churchill Lectures in Economics was inaugurated in 1993 to provide a se- ries of public lectures on topics of current interest to students and researchers in the discipline. The lectures will be selected from the top echelon of leading scholars in the profession. Although they will always be acknowledged special- ists in their field, they will be encouraged to take a broad look at their chosen subject and to reflect in a way that will be accessible to senior undergraduates and graduate students. Peter Diamond, On Time, 1994 Douglas Gale, Strategic Foundations of General Equilibrium: Dynamic Match- ing and Bargaining Games, 2000 Ariel Rubinstein, Economics and Language, 2000 iii P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610-FM CB610-Milgrom-v3 October 27, 2003 16:27 iv P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610-FM CB610-Milgrom-v3 October 27, 2003 16:27 PUTTING AUCTION THEORY TO WORK PAUL MILGROM Stanford University v cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521551847 © Paul Milgrom 2004 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2004 isbn-13 978-0-511-16612-9 eBook (NetLibrary) isbn-10 0-511-16612-5 eBook (NetLibrary) isbn-13 978-0-521-55184-7 hardback isbn-10 0-521-55184-6 hardback isbn-13 978-0-521-53672-1 paperback isbn-10 0-521-53672-3 paperback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610-FM CB610-Milgrom-v3 October 27, 2003 16:27 Contents Preface page xi Foreword by Evan Kwerel xv 1 Getting to Work 1 1.1 Politics Sets the Stage3 1.2 Designing for Multiple Goals3 1.2.1 Substitutes and Complements6 1.2.2 New Zealand’s Rights Auction9 1.2.3 Better Auction Designs 13 1.2.4 The FCC Design and Its Progeny 13 1.3 Comparing Seller Revenues 16 1.4 The Academic Critics 19 1.4.1 Resale and the Coase Theorem 19 1.4.2 Mechanism Design Theory 21 1.4.3 Theory and Experiment 25 1.4.4 Practical Concerns 26 1.5 Plan for This Book 31 PART I THE MECHANISM DESIGN APPROACH 35 2 Vickrey–Clarke–Groves Mechanisms 45 2.1 Formulation 45 2.2 Always Optimal and Weakly Dominant Strategies 49 2.3 Balancing the Budget 53 2.4 Uniqueness 55 2.5 Disadvantages of the Vickrey Auction 56 2.5.1 Practical Disadvantages 56 2.5.2 Monotonicity Problems 57 2.5.3 The Merger–Investment Disadvantage 60 2.6 Conclusion 61 vii P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610-FM CB610-Milgrom-v3 October 27, 2003 16:27 viii Contents 3 The Envelope Theorem and Payoff Equivalence 64 3.1 Hotelling’s Lemma 65 3.2 The Envelope Theorem in Integral Form 66 3.3 Quasi-linear Payoffs 69 3.3.1 Holmstrom’s Lemma 70 3.3.2 The Green–Laffont–Holmstrom Theorem 71 3.3.3 Myerson’s Lemma 73 3.3.4 Revenue Equivalence Theorems 75 3.3.5 The Myerson–Satterthwaite Theorem 77 3.3.6 The Jehiel–Moldovanu Impossibility Theorems 80 3.3.7 Myerson and Riley–Samuelson Revenue-Maximizing Auctions 84 3.3.8 The McAfee–McMillan Weak-Cartels Theorem 87 3.3.9 Sequential Auctions and Weber’s Martingale Theorem 90 3.3.10 Matthews Theorem: Risk Averse Payoff Equivalence 91 3.4 Conclusion 94 4 Bidding Equilibrium and Revenue Differences 98 4.1 The Single Crossing Conditions 99 4.1.1 The Monotonic Selection Theorem 101 4.1.2 The Sufficiency Theorem 102 4.1.3 The Constraint Simplification Theorem 105 4.1.4 The Mirrlees–Spence Representation Theorem 106 4.2 Deriving and Verifying Equilibrium Strategies 110 4.2.1 The Second-Price Auction with a Reserve Price 111 4.2.2 The Sealed Tender, or First-Price, Auction 112 4.2.3 The War of Attrition Auction 117 4.2.4 The All-Pay Auction 119 4.3 Revenue Comparisons in the Benchmark Model 119 4.3.1 Payoff Equivalence without Revenue Equivalence 121 4.3.2 Budget Constraints 132 4.3.3 Endogenous Quantities 135 4.3.4 Correlated Types 137 4.4 Expected-Revenue-Maximizing Auctions 140 4.4.1 Myerson’s Theorem 144 4.4.2 Bulow–Klemperer Theorem 148 4.4.3 The Irregular Case 148 4.5 Auctions with Weak and Strong Bidders 149 4.6 Conclusion 154 P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610-FM CB610-Milgrom-v3 October 27, 2003 16:27 Contents ix 5 Interdependence of Types and Values 157 5.1 Which Models and Assumptions are “Useful”? 158 5.1.1 Payoffs Depend Only on Bids and Types 158 5.1.2 Types Are One-Dimensional and Values Are Private 159 5.1.3 Types Are Statistically Independent 161 5.2 Statistical Dependence and Revenue-Maximizing Auctions 162 5.3 Wilson’s Drainage Tract Model 166 5.3.1 Equilibrium 167 5.3.2 Profits and Revenues 173 5.3.3 Bidder Information Policy 175 5.3.4 Seller Information Policy 177 5.4 Correlated Types and Interdependent Values 181 5.4.1 Affiliation 182 5.4.2 The Milgrom–Weber Ascending Auction Models 187 5.4.2.1 The (Second-Price) Button Auction with Minimal Information 188 5.4.2.2 The Button Auction with Maximal Information 195 5.4.2.3 Some Revenue Comparisons 198 5.4.3 First-Price Auctions 200 5.5 Conclusion 204 6 Auctions in Context 208 6.1 The Profit and Surplus Contribution of an Entrant 214 6.2 Symmetric Models with Costly Entry 216 6.2.1 Symmetric Bidders and Uncoordinated Entry 218 6.2.1.1 Equilibrium in Entry and Bidding Decisions 218 6.2.1.2 Setting the Reserve Price 222 6.2.2 Coordinating Entry among Symmetric Competitors 225 6.2.2.1 Pre-qualifying Bidders 227 6.2.2.2 Auctions, Negotiations, and Posted Prices 230 6.2.2.3 Buy Prices 232 6.3 Asymmetric Models: Devices to Promote Competition 234 6.3.1 Example of Set-asides 235 6.3.2 Example of Bidding Credits 237 6.3.3 Example of Lot Structure and Consolation Prizes 238 6.3.4 Premium Auctions 239 6.3.5 Dutch vs.
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